Le Gall [1]}, based on [2]}, proved that if, loosely speaking, the weight sequence \({q}\)  is such that the degree of a typical face of \(M^{{q}}\)  has finite exponential moments, then \(M^{{q}}\) , conditioned to have \(n\)  vertices, and rescaled by a factor of order \(n^{-1/4}\)  converges in distribution to the Brownian map.
On the other hand, when the weight sequence \({q}\)  is such that, roughly speaking, a typical face of \(M^{{q}}\)  has degree \(2k\)  with probability of order \(c k^{-1-\alpha }\)  when \(k \rightarrow \infty \) , with \(c>0\)  and \(\alpha \in (1,2)\) , then Le Gall & Miermont [3]} proved a tightness result: From every sequence of integers, one can extract a subsequence along which the map \(M^{{q}}\)  conditioned to have \(n\)  vertices, rescaled by a factor \(n^{-1/(2\alpha )}\) , converges in distribution towards a limit space which has almost surely Hausdorff dimension \(2\alpha \) .
Both results were extended in [4]} to the full domain of attraction of a stable law with index in \((1,2]\) , and by conditioning with respect to either the number vertices, or of edges, or of faces.
